Gas sponges capable of absorbing, storing, and releasing ions in a reversible manner are in high demand for advanced electronics, energy devices, and sensors. Here, it is shown that brownmillerite BaInO2.5 epitaxial films exhibit the capability to act as solid‐state catalytic hydrogen sponges at a remarkably low temperature (≈100 °C). Compared to sintered pellets with random crystallographic orientations and many defects, BaInO2.5 epitaxial films give three orders of magnitude higher conductivity of hydrogen ions, which can be linked to the presence of well‐ordered 1D channels and lack of grain boundaries and other defects. Using scanning transmission electron microscopy, it is shown that hydrogen ions can be stored near InO4 tetrahedral layers of brownmillerite without destroying the parent framework. The high performance and reproducibility of the BaInO2.5 epitaxial films coupled with ultralow power consumption make them ideal candidates for neuromorphic devices and beyond.